Hitting Matrix and Domino Tiling with Diagonal Impurities

As a continuation to our previous work (Nakano and Sadahiro in Fundam. Inform. 117:249-264, 2012; Nakano and Sadahiro in J. Stat. Phys. 139(4):565-597, 2010), we consider the domino tiling problem with impurities. (1) If we have more than two impurities on the boundary, we can compute the number of corresponding perfect matchings by using the hitting matrix method (Fomin in Trans. Am. Math. Soc. 353(9):3563-3583, 2001). (2) We have an alternative proof of the main result in Nakano and Sadahiro (Fundam. Inform. 117:249-264, 2012) and result in (1) above using the formula by Kenyon and Wilson (Trans. Am. Math. Soc. 363(3):1325-1364, 2011; Electron. J. Comb. 16(1):112, 2009) of counting the number of groves on circular planar graphs. (3) We study the behavior of the probability of finding the impurity at a given site when the size of the graph tends to infinity, as well as the scaling limit of those.